Answering counterfactual queries has many important applications such as knowledge discovery and explainability, but is challenging when causal variables are unobserved and we only see a projection onto an observation space, for instance, image pixels. One approach is to recover the latent Structural Causal Model (SCM), but this typically needs unrealistic assumptions, such as linearity of the causal mechanisms. Another approach is to use na\"ive ML approximations, such as generative models, to generate counterfactual samples; however, these lack guarantees of accuracy. In this work, we strive to strike a balance between practicality and theoretical guarantees by focusing on a specific type of causal query called domain counterfactuals, which hypothesizes what a sample would have looked like if it had been generated in a different domain (or environment). Concretely, by only assuming invertibility, sparse domain interventions and access to observational data from different domains, we aim to improve domain counterfactual estimation both theoretically and practically with less restrictive assumptions. We define domain counterfactually equivalent models and prove necessary and sufficient properties for equivalent models that provide a tight characterization of the domain counterfactual equivalence classes. Building upon this result, we prove that every equivalence class contains a model where all intervened variables are at the end when topologically sorted by the causal DAG. This surprising result suggests that a model design that only allows intervention in the last $k$ latent variables may improve model estimation for counterfactuals. We then test this model design on extensive simulated and image-based experiments which show the sparse canonical model indeed improves counterfactual estimation over baseline non-sparse models.

翻译：回答反事实查询具有知识发现和可解释性等许多重要应用，但当因果变量未被观测且仅能观察到其在观测空间（例如图像像素）上的投影时，这一任务极具挑战性。一种方法是恢复潜在结构因果模型，但这通常需要不切实际的假设，例如因果机制的线性。另一种方法是使用朴素的机器学习近似方法（如生成模型）生成反事实样本，但这些方法缺乏准确性保证。本文致力于在实用性与理论保障之间取得平衡，重点关注一类特定因果查询——领域反事实，它假设样本在不同领域（或环境）中生成时可能呈现的样貌。具体而言，我们仅假设可逆性、稀疏领域干预以及可获取不同领域的观测数据，旨在以更宽松的假设从理论和实践层面改进领域反事实估计。我们定义了领域反事实等价模型，并证明了等价模型的充要性质，从而严格刻画了领域反事实等价类。基于这一结果，我们证明了每个等价类中存在一个模型，其中所有干预变量按因果有向无环图拓扑排序后位于末端。这一惊人结论表明，仅允许对最后k个潜变量进行干预的模型设计可能改善反事实的模型估计。随后，我们通过大量模拟和基于图像的实验验证了该模型设计，结果表明稀疏规范模型确实比基线非稀疏模型更能提升反事实估计效果。